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Combining unbiased estimators[edit]

Let and be unbiased estimators of with non-singular variances and respectively.

Then the minimum variance linear unbiased estimator of is obtained by combining and using weights that are proportional to the inverses of their variances. The result can be expressed in a variety of ways:

The proof is an application of the principle of Generalized Least-Squares. The problem can be formulated as a GLS problem by considering that: with

Applying the GLS formula yields:


Expected value of SSH[edit]

Consider one-way MANOVA with groups, each with observations. Let and let

be the design matrix.

Let be the residual projection matrix defined by

Analyzing SSH[edit]

We can find expressions for SSH in terms of the data and find expected values for SSH under a fixed effects or under a random effects model.

The following formula is used repeatedly to find the expected value of a quadratic form. If is a random vector with and , and is symmetric, then

We can model:



and is independent of .




where is the group-size weighted mean of group sizes. With equal groups and



Multivariate response[edit]

If we are sampling from a p-variate distribution in which


then the analogous results are:


Note that

and that the group-size weighted average of these variances is:

The expectation of combinations of and of the form :

1 0
0 1